In posts 20.2, 20.3 and 20.5, we thought about stretching a long, thin wire. This post continues the topic. But it is about the work done (post 16.20) in stretching the wire and what happens to the resulting potential energy (post 16.21).
When a force stretches the wire, it moves its point of application and so does work (post 16.20). This work may be stored as potential energy (post 16.21). If all the work done is stored, the potential energy can be used to restore the wire to its return to its original dimensions, when the force is removed. This return to the original dimensions is called recoil – the wire acts in the same way as a spring (post 16.49). If, the wire has a constant stiffness, k, (which means that it obeys Hooke’s law, post 20.3) the work done is equal to
W = ½ F(ΔL)2
where F is the magnitude of the force and ΔL is the extension of the wire (post 16.49).
The ability to store the work done on the wire as potential energy, that is available for recoil, is called elasticity. Then the material of the wire is elastic. Unfortunately, the words “elastic” and elasticity” are often incorrectly used, even in the scientific literature; this is important because effective scientific communication depends on the precise use of technical words (post 17.14). You will sometimes see these words used to mean that something is easy to stretch – the correct terms are compliance and compliant – rubber, for example, has a high compliance and is a compliant material.
In general, when a force F moves its point of application from 0 to a distance ΔL, the work done (post 17.36) is
In our example, the wire stretches in the direction of the applied force. We define this direction by the unit vector i, so that
F = iF and dr = idr.
Here symbols in bold are vectors and symbols in normal font are scalars – this is explained in post 17.2. Then
F.dr = (i.i)Fdr = Fdr.
This is explained in appendix 2 of post 17.13.
So, in general the work done by the force, in stretching the wire, is given by
which is the area under the graph shown above (see post 17.19). In an elastic material this work is stored as potential energy that is used in recoil. Since the magnitude of the potential energy used in recoil is equal to the magnitude of the work done, the wire follows the same path on the graph, of force plotted against deformation, as it did when it was stretched.
The property of following the same path during stretching and recoil is characteristic of elastic materials.
From the definition of stress, σ, and strain, ε, post 20.2,
where A is the cross-sectional area of the wire, L is its length and V is its volume. So the area under a stress-strain curve (see the pictures in post 20.3) is the energy given to a unit volume of the material of the wire.
Now let’s suppose the material of our wire obeys Hooke’s law (post 20.3). Then
which is the same result that was obtained, by a less rigorous method, for a spring in post 16.49. If you’re not sure how this definite integral was evaluated, see post 17.19.
This post comes with a warning. Many textbooks give the impression that all solids are elastic materials – they’re not!
20.5 Poisson’s ratio
20.3 Hooke’s law
20.2 Deformation of objects
16.37 Solids, liquids and gases
20.9 Stiffness and strength
20.11 Elasticity of rubber